Ext⁰(M, N) = Hom(M, N): The Ultimate Proof

Hey guys! Ever wondered how the Ext functor and the Hom functor cozy up together, especially when we're talking about Ext⁰? Well, you're in the right place! Today, we're diving deep into proving that Ext⁰(M, N) is actually the same as Hom(M, N). This is a fundamental concept in homological algebra, and trust me, understanding it opens up a whole new world of algebraic structures. We'll break it down step by step, making sure even if you're just starting out with these ideas, you'll get the hang of it. So, let's roll up our sleeves and get started!

What's the Big Deal with Ext and Hom?

Before we jump into the proof, let's quickly recap what Ext and Hom are all about. Think of the Hom functor, Hom(M, N), as a way to look at all the cool maps (or homomorphisms) that take you from one module M to another module N. It's like a collection of all the possible ways M can be transformed into N while preserving the algebraic structure. Now, the Ext functor is a bit more sophisticated. It measures how far away a module is from being projective (we'll get to what that means soon!). In simpler terms, Ext helps us understand the 'extensions' of modules, which are ways of building new modules from old ones.

The connection between Ext and Hom is super important because it bridges the gap between simple mappings and more complex algebraic relationships. When we say Ext⁰(M, N) = Hom(M, N), we're saying that the 'zeroth extension' between M and N is precisely the set of all homomorphisms from M to N. This is a foundational result, and it's used all over the place in homological algebra. To truly grasp this, we need to delve into projective resolutions and how they play a role in defining Ext. Imagine you have a module M. A projective resolution of M is like a super-detailed roadmap that breaks M down into a sequence of projective modules (modules with special lifting properties). These projective modules, denoted as Pᵢ, are connected by homomorphisms, creating a chain complex. This chain complex allows us to study the properties of M in a more structured way. The beauty of projective resolutions lies in their ability to simplify complex module structures into manageable, projective components. By understanding how M can be constructed from these projectives, we gain deeper insights into its nature and behavior. Projective resolutions are not unique; a module can have several different projective resolutions. However, the magic lies in the fact that regardless of the chosen resolution, the Ext functor remains consistent. This consistency is crucial because it ensures that Ext provides a reliable measure of the module's extension properties. Think of it like having different routes to the same destination – each route might be different, but the destination remains the same. This invariance of Ext with respect to the choice of resolution is a powerful tool that allows us to choose the most convenient resolution for a given problem. In essence, projective resolutions serve as a lens through which we examine modules, revealing their hidden structures and relationships. They form the backbone of homological algebra, enabling us to define and compute Ext functors, which in turn provide a wealth of information about module extensions and their properties. So, next time you encounter a projective resolution, remember it's not just a sequence of modules and homomorphisms; it's a key to unlocking the secrets of algebraic structures.

Setting the Stage: Projective Resolutions

Okay, so what's a projective resolution? It's a sequence of projective modules (modules that are